Method and apparatus for compressing and expanding image data

ABSTRACT

An image compression apparatus transforms original image data partitioned into first blocks, each of which is composed of a plurality of pixels, to reduced-image data composed of a smaller number of pixels than that of the original image data. Further, the apparatus generates expanded-image data from the reduced-image data by fluency transform. Based on the expanded-image data and the original image data, differential value data is obtained, and the differential value is transformed to differential DCT coefficient data by DCT processing. Based on the differential DCT coefficient data, a code-length corresponding to a bit-length necessary for a Huffman coding is calculated. The fluency transform has a plurality of modes, each mode being selected in order, the code-length being calculated for all of the modes. Then, a mode which makes the code-length minimum is determined as the optimum mode. The differential DCT coefficient data is Huffman encoded, so that Huffman-encoded bit data is generated. Then, in an image expansion apparatus, the original image data is restored on using the optimum mode, reduced-image data and the Huffman-encoded bit data.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a method and apparatus for compressing andexpanding digital image data, and especially relates to imagecompression by reducing the number of pixels of original image data, andimage expansion by increasing the number of pixels of reduced-imagedata.

2. Description of the Related Art

In an example of an image compression method, it has been known tocalculate the average value of a predetermined number of pixels. In thiscompression method, applied to original digital image data, which ispartitioned into pixel-blocks composed of a plurality of pixels, anaverage pixel value for the plurality of pixels is calculated in eachblock. Consequently, reduced-image data composed of pixels having theaverage pixel values is obtained. When expanding the reduced-image datato restore the original image, an interpolation processing, such as alinear interpolation, is usually performed so that expanded-image datacorresponding to the original image data is obtained.

However, since part of the information included in the original imagedata is lost in the process of generating the reduced-image data, pixelvalues generated by the interpolation processing are not necessarilyequal to corresponding pixel values in the original image data. Namely,the expanded-image data does not coincide with the original image-data.Therefore, picture quality decreases in the process of the compressionand expansion processing, and the original image data can not becompletely restored.

SUMMARY OF THE INVENTION

Therefore, an object of the present invention is to provide a method andapparatus for compressing and expanding digital image data efficiently,while limiting degradation in picture quality.

The compression apparatus according to the present invention has areduced-image generating processor, a fluency transform processor, adifferential value calculating processor, an orthogonal transformprocessor, a mode setting processor, a code-length calculatingprocessor, an optimum mode determining processor and an optimum Entropycoding processor. The reduced-image generating processor transformsoriginal imaged at a partitioned into first blocks, each of which iscomposed of a plurality of pixels, to reduced-image data composed of asmaller number of pixels than that of the original image data. Thefluency transform processor applies a fluency transform to thereduced-image data so as to generate expanded-image data partitionedinto second blocks corresponding to the first blocks. Note that, thefluency transform has a plurality of modes. The differential valuecalculating processor obtains differential value data indicating adifference between the original image data and the expanded-image data.The orthogonal transform processor obtains orthogonal transformcoefficient data by applying an orthogonal transform to the differentialvalue data. The mode setting processor selects one mode from theplurality of modes. Thus, the orthogonal transform coefficient data isgenerated in accordance with the selected mode. The code-lengthcalculating processor calculates a code-length corresponding to a bitlength of an Entropy-encoded bit data obtained by Entropy coding of theorthogonal transform coefficient data. The code-length calculatingprocessor calculates the code-length in each of the plurality of modes.The optimum mode determining processor determines an optimum mode, bywhich the code-length becomes minimum, from the plurality of modes. Theoptimum Entropy coding processor obtains Entropy-encoded bit data byapplying the Entropy coding to the orthogonal transform coefficient datain accordance with the optimum mode. Preferably, the compressionapparatus has a recording medium for recording the reduced-image data,the optimum mode and the Entropy-encoded bit data.

On the other hand, an expansion apparatus according to the presentinvention has a data reading processor, an optimum mode settingprocessor, an expanded-image generating processor, an Entropy decodingprocessor, an inverse orthogonal transform processor and an originalimage data restoring processor. The data reading processor reads thereduced-image data, the Entropy-encoded bit data and the optimum moderecorded in the recording medium. The optimum mode setting processorsets the optimum mode from the plurality of modes.

The expanded-image generating processor applies the fluency transformaccording to the optimum mode to the reduced-image data so that theexpanded-image data is obtained. The Entropy decoding processor thatrestores the orthogonal transform coefficient data by applyingEntropy-decoding to the Entropy-encoded bit data. The inverse orthogonaltransform processor restores the differential value data by applying aninverse orthogonal transform to the orthogonal transform coefficientdata. The original image data restoring processor restores the originalimage data on the basis of the expanded-image data and the differentialvalue data.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be better understood from the description ofthe preferred embodiments of the invention set forth below together withthe accompanying drawings, in which:

FIG. 1 is a block diagram of an image compression apparatus of anembodiment.

FIG. 2 is a block diagram of an image expansion apparatus of anembodiment.

FIG. 3 is a view showing an image reduction of original image data.

FIG. 4 is a view showing the expansion processing.

FIG. 5 is a view showing an example of a matrix arrangement ofdifferential values and a matrix arrangement of differential DCTcoefficients.

FIG. 6 is a view showing an example of Huffman coding.

FIGS. 7A, 7B, 7C and 7D are views showing fluency functions varying withparameter m.

FIG. 8 is a normalized fluency function at the parameter m=1.

FIG. 9 is a view showing a fluency transform processing.

FIG. 10 is a view showing the fluency transform along a horizontaldirection.

FIG. 11 is a view showing a table T1 indicating 8 pixel values along thehorizontal direction in a block.

FIG. 12 is a view showing the fluency transform along a verticaldirection.

FIG. 13 is a view showing a table T2 representing 8 pixel values alongthe vertical direction in the block.

FIG. 14 is a view showing a flowchart of the process for determining theoptimum mode.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, the preferred embodiments of the present invention aredescribed with reference to the attached drawings.

FIGS. 1 and 2 are block diagrams of an image compression and expansionapparatus according to an embodiment of the present invention. Notethat, the image compression and expansion apparatus of the embodimentare incorporated in a digital still camera, and a CPU (not shown) in thecamera controls the image compression and expansion apparatus.

Light, reflected on a subject S to be photographed, passes through aphotographing optical system L, whereby the subject image is formed on alight-receiving area of a CCD 31. On the light-receiving area, red (R),green (G), blue (B) color filters are provided, thus analog image-pixelsignals corresponding to the object image are generated in the CCD 31.The analog image-pixel signals are converted to the digital imagesignals in a A/D converter 32, and then are subjected to an imageprocessing in an image processing circuit (not shown), so that luminancedata Y and color difference data Cb,Cr are generated. The luminance dataY and color difference data Cb,Cr are temporally stored in a memory 33.The memory 33 is divided into independent memory areas for storing theluminance data Y and color difference data Cb,Cr separately. Each memoryarea has a storage capacity of one frame worth of the subject image.

The luminance data Y and color difference data Cb,Cr stored in thememory 33 are fed to an image compression apparatus 10. The imagecompression apparatus 10 has a reduced-image generating processor 11, amode setting processor 12, a fluency transform processor 13, adifferential value calculating processor 14, a DCT (Discrete CosineTransform) processor 15, a code-length calculating processor 16, anoptimum mode determining processor 17, a Huffman coding processor 18 anda recording medium M. The luminance data Y and color difference dataCb,Cr are subject to compression processing in the image compressionapparatus 10, separately.

The original image data is partitioned into a plurality of pixel-blocks,and each pixel-block of the luminance data Y and color difference dataCb,Cr are separately subjected to a reducing processing in thereduced-image generating processor 11. Thus, reduced-image data,composed of a smaller number of pixels than that of the original imagedata, is generated. The reduced-image data is stored in a recording areaM1 of the recording medium M. Herein, the recording medium M is an ICmemory card.

In this embodiment, as described later, a fluency transform having aplurality of modes is executed as an expansion processing in the imageexpansion apparatus 20. However, expanded-image data obtained by theexpansion processing is not identical with the original image data.Therefore, in the image compression apparatus 10, the fluency transformis executed, differential value data indicating a difference between theexpanded-image data obtained by the fluency transform and the originalimage data is obtained, and the difference value data is encoded. Forencoding, a JPEG (Joint Photographic Experts Group), which is astandardized compression method of still image data, is applied.

In the mode setting processor 12, one mode is selected from theplurality of modes, and then the reduced-image data is subjected to thefluency transform with the selected mode, by the fluency transformprocessor 13. Thus, expanded-image data corresponding to the originalimage data is obtained. The number of pixels of the expanded-image datacoincides with that of the original image data. In the differentialvalue calculating processor 14, the differential value data iscalculated.

In the DCT processor 15, the differential value data is subjected to DCTprocessing, so that the differential value data is transformed todifferential DCT coefficient data. The DCT is an orthogonal transform,therefore the differential value data is transformed into orthogonaltransform coefficient data. In a code-length calculating processor 16, acode-length for Huffman-coding is calculated. Namely, a bit length ofHuffman-encoded bit data, obtained by the Huffman coding of thedifferential DCT coefficient data, is calculated as described later.

After the code-length for the selected mode is calculated, the next modeis selected from the other modes by the mode setting processor 12. Then,the reduced-image data is subjected to the fluency transform using thenewly selected mode in the fluency transform processor 13. Thecode-length is calculated in the code-length calculating processor 16.Hence, the code-length is calculated with regard to all of the modes inorder.

In the optimum mode determining processor 17, the optimum mode isdetermined from all of the modes. When the reduced-image data issubjected to the fluency transform depending upon the optimum mode, thecode-length becomes a minimum. The optimum mode is recorded in arecording area M2 of the recording medium M.

After the fluency transform with the optimum mode is applied to thereduced-image data and the generated expanded-image data by fluencytransform is transformed to the differential DCT coefficient data, viathe differential value calculating processor 14 and the DCT processor15, the differential DCT coefficient data is subjected to Huffman codingin the Huffman coding processor 18. Consequently, Huffman-encoded bitdata in accordance with the optimum mode is generated. The encoded bitdata is recorded in a recording area M3 of the recording medium M. Notethat, the Huffman coding is Entropy coding.

As shown in FIG. 2, the image expansion apparatus 20 includes a optimummode setting processor 21, an expanded-image generating processor 22, aHuffman decoding processor 23, an inverse DCT processor 24 and a pixelvalue synthesizing processor 25. The reduced-image data, the optimummode and the encoded bit data, obtained in the image compressionapparatus 10, are recorded in the recording medium M separately.

When the optimum mode is read from the recording area M2, the mode ofthe fluency transform for expanding the reduced-image data is set to theoptimum mode in the optimum mode setting processor 21. Then, thereduced-image data read from the recording area M1 is subjected to thefluency transform in the expanded-image generating processor 22, thusthe expanded-image data corresponding to the original image data isgenerated.

In the Huffman decoding processor 23, the encoded bit data read from therecording area M3 is subjected to a Huffman decoding so that thedifferential DCT coefficient data is restored. The Huffman decoding isan Entropy decoding, and is an inverse of the Huffman coding. In theIDCT processor 24, the differential DCT coefficient data is subjected tothe IDCT processing, so that the differential value data is restored.The IDCT processing is an inverse orthogonal transform, and is aninverse of the DCT processing.

In the pixel value synthesizing processor 25, the differential valuedata obtained by the IDCT processor 24 is subtracted from theexpanded-image data obtained by the expanded-image generating processor22. Consequently, the original image data is restored. The originalimage data is temporally stored in the memory 33, and fed to a display(not shown), such as a LCD (Liquid Crystal Display). Consequently, thephotographed image is shown on the display. Note that, the compressionand expansion processing in the embodiment is substantially a losslesscoding and decoding, whereby no part of the original image data is lost.

FIG. 3 is a view showing an image reduction processing to the originalimage data.

The original image data, represented by “P”, has 1024×512 pixels, and ispartitioned into a plurality of blocks B, each of which is composed of8×8(=64) pixels P_(vu). The plurality of blocks is arranged in a matrix.A range of value of each pixel P_(vu) is 0 to 255, and pixel values arealso represented by “P_(vu)”. As shown in FIG. 3, u-v coordinates aredefined with respect to the original image data P. The left corner ofthe original image data P is set as the origin, u-axis is parallel tothe horizontal direction (width direction) and v-axis is parallel to thevertical direction (length direction).

In the reduced-image generating processor 11 (shown in FIG. 1), anaverage value of the 8×8 pixels P_(vu) is calculated in each of theblocks B. Consequently, the reduced-image data R composed of averagedpixels, each value being the average value of each of the blocks B, isgenerated. For example, as shown in FIG. 3, a block B0, positioned atthe s-th block along the u-axis and the t-th block along the v-axis, istransformed to a pixel R_(ts) of the reduced-image data R on the basisof the following formula. $\begin{matrix}{R_{ts} = {\left( {\sum\limits_{u = {s \times 8}}^{{s \times 8} + 7}\quad {\sum\limits_{v = {t \times 8}}^{{t \times 8} + 7}P_{vu}}} \right)/64}} & (1)\end{matrix}$

The pixel R_(ts) is positioned at the s-th pixel along the horizontaldirection and the t-th pixel along the vertical direction in thereduced-image data R.

In this way, the reduced-image data R is obtained by calculating theaverage value of each block B. Namely, the original image data P iscompressed. The pixel number of the reduced-image data R is {fraction(1/64)} of that of the original image data P.

FIG. 4 is a view showing the expansion process, executed in theexpanded-image generating processor 22 in the image expansion apparatus20 shown in FIG. 2, and further in the fluency transform processor 13 ofthe image compression apparatus 10 shown in FIG. 1.

Each pixel of the reduced-image data R is firstly subjected to thefluency transform along the horizontal direction and is secondlysubjected to the fluency transform along the vertical direction, so thatthe blocks B′ composed of the 8×8(=64) pixels are generated. Forexample, when the pixel R_(ts) is subjected to the fluency transform,firstly, a block BP composed of 8 pixels in the horizontal direction isgenerated. Then, based on the 8 pixels of the block BP, the block B0′composed of 8×8 pixels is generated. The block B0′ corresponds to theblock B0 in the original image data P.

When all pixels of the reduced-image data R are subjected to the fluencytransform, the expanded-image data J partitioned into blocks B′ composedof 8×8 pixels (J_(vu)) is generated. The number of blocks B′ in theexpanded-image data J is equal to that of the original image data P.

Herein, pixels in the block B0′ that are expressed by “I′_(yx)(0≦x≦7,0≦y≦7)”, and the pixels I′_(y x)(0≦x≦7,0 ≦y≦7)” satisfy thefollowing formula. Note that, x-y coordinates are defined with respectto the original image data P, the x-axis is parallel to the horizontaldirection and the y-axis is parallel to the vertical direction.$\begin{matrix}{J_{{{t \times 8} + y},{{s \times 8} + x}} = I_{yx}^{\prime {({s,t})}}} & (2)\end{matrix}$

In this way, the reduced-image data R is transformed to theexpanded-image data J. Namely, the reduced-image data R is expanded.

FIG. 5 is a view showing a process, executed in the differential valuecalculating processor 14 and DCT processor 15. Then, FIG. 6 is a viewshowing a process executed in the Huffman coding processor 18. Herein,as an example, the block B0′ is subjected to the Huffman coding.

In FIG. 5, the block B0, the block B0′, a differential value matrixrepresented by “D” and a differential DCT coefficient matrix representedby “C” is shown. The block B0, B0′ are composed of 8×8 pixels P_(yx) andI′_(yx) respectively. Further, the differential value matrix D iscomposed of 8×8 differential values D_(yx) and the differential DCTmatrix C_(ji) is composed of 8×8 differential DCT coefficients. Notethat, suffix “j” indicates the vertical direction similar to the y-axis,suffix “i” indicates the horizontal direction similar to the x-axis. Forexample, when i is “1” and j is “1”, the differential DCT coefficientC₁₁ is “1” as shown in FIG. 5.

In the first instance, the difference values D_(yx), indicating adifference between the pixels P_(yx) and the corresponding pixelsI′_(yx), is obtained by

D _(yx) =I′ _(yx) −P _(yx)  (3)

For example, substituting a pixel P_(1O) (=74) and I′₁₀ (=77) for theformula (3), a differential value D₁₀ (=3) is obtained.

Then, the differential values D_(yx) are converted to 64 (=8×8)differential DCT coefficients C_(ji) by the DCT processing. Thedifferential DCT coefficient C₀₀ (=−2) at position (0,0) is a DC (DirectCurrent) component, while the remaining 63 differential DCT coefficientsC_(ji) are AC (Alternating Current) components. The AC components showhow many higher spatial frequency components there are in thedifferential DCT coefficient matrix C from the differential coefficientC₁₀ or C₀₁ to the differential DCT coefficient C₇₇. The DC componentshows an average value of the spatial frequency components of the 8×8differential values D_(yx) as a whole.

In the second instance, the Huffman coding is performed to thedifferential DCT coefficient matrix as follows. Note that, as the DCcomponent, or the differential DCT coefficient C₀₀ (=−2) is close to“0”, the Huffman coding for the AC components is applied to thedifferential DCT coefficient C₀₀. In other words, the Huffman coding forthe DC component is not applied.

As shown in FIG. 6, the differential DCT coefficients C_(ji) are zigzagscanned along the arrow direction, and are rearranged into aone-dimensional array. In the array (−2, −20, −18, 9, 0, . . . . . . ),the differential DCT coefficients, the value of which are not “0”, areclassified by using a categorized table (not shown), thus additionalbits are obtained on the basis of corresponding category numbers. On theother hand, when the differential DCT coefficient C_(ji) is “0”, anumber of consecutive differential DCT coefficients equal to “0” arecounted in the one-dimensional array, thus a length of consecutive “0s”,namely, a zero run length Z is obtained. Encoded bit data is thenobtained by combining the category number H and the zero run length Z.

As an example, in the case of the differential DCT coefficient C₁₀(=−18), the category number H is “5”, thus the additional bits “01101”are obtained. As no differential DCT coefficient of “0” exist in frontand behind, the zero run length Z is “0”. Therefore, the category numberH (=5) and the zero run length Z (=0) is represented by “05” as shown inFIG. 6. Then, based on the zero run length Z (=0) and the categorynumber H (=5), a Huffman table (not shown) is referenced to, so thatcode-word “11010” is obtained. by combining the additional bits “01101”and the codeword “11010”, the encoded bit data “1101001101” is obtained.In the same way, other differential DCT coefficients C_(ji) are subjectto the Huffman coding, so that one block worth of the encoded bit datais obtained.

When the Huffman coding is applied to one frame worth of the objectimage, the encoded bit data is recorded in the recording area M3. TheHuffman table and the table for categorizing are default tables, whichare used in a conventional JPEG Baseline algorithm.

In the image expansion apparatus 20, the Huffman decoding and the IDCTprocessing, which are the inverse of the Huffman coding and the DCTprocessing shown in FIGS. 5 and 6 respectively, are executed. Then, asdescribed above, based on the differential value and the expanded-imagedata J, the original image data P is obtained. For example, regardingthe blocks B0, B0′, the original image data P_(yx) is obtained by

P _(yx) =I′ _(yx) −D _(yx)  (4)

Hereinafter, with reference to FIGS .7 to 13, the fluency transform isexplained. Since the fluency transform is based on a fluency function,the fluency function will be described before the explanation of thefluency transform.

The fluency function, named by professors Kamada and Toraichi, is knownas a function, which can represent various signals appropriately, forexample, disclosed in a Mathematical Physics Journal (SURI-KAGAKU) No.363, pp 8-12 (1993), published in Japan.

To begin with, a fluency function space is defined as follows:

It is supposed that a function space composed of a staircase (scaling)function, which is obtained on the basis of a rectangular functionrepresented by formula (5), is represented as shown in the followingformula (6), the fluency function space is defined by formula (7).$\begin{matrix}{{\chi (t)} = \left\{ \begin{matrix}{1,} & {{t} \leq {1/2}} \\{0,} & {otherwise}\end{matrix} \right.} & (5) \\{\quad^{1}S:=\left\{ {{{f:\left. \left. R\rightarrow C \right. \middle| {f(t)} \right.} = {\sum\limits_{n = {- \infty}}^{\infty}\quad {b_{n}{\chi \left( {t - n} \right)}}}},{\left\{ b_{n} \right\} \in l^{2}}} \right\}} & (6) \\{{\,^{m}S}:={\left\{ {{{h:\left. \left. R\rightarrow C \right. \middle| {h(t)} \right.} = {\int_{- \infty}^{\infty}{{f\left( {t - \tau} \right)}{g(\tau)}\quad {\tau}}}},\quad {f \in^{m - 1}S},\quad {g \in^{1}S}} \right\} \left( {m \geq 2} \right)}} & (7)\end{matrix}$

The fluency function space of order m “^(m)S” is constructed by afunction system, composed of (m−2)-times continuously differentiablepiecewise polynomial of degree m−1. The fluency function is a staircase(scaling) function when m is 1. The formula (5), which representsrectangular function, indicates a sampling basis at order m=1, and theformula (6) indicates the fluency function space at order m=1. Thefluency function space ^(m)S is a series of a function space, which canconnect from a staircase function space (m=1) to a Fourier band-limitedfunction space (m=∞). Note that, a continuous differentiability m isregarded as a parameter.

The function system, characterizing the fluency function space ^(m)S andcorresponding to an impulse response, is derived from a biorthogonalsampling basis theorem composed of a sampling basis and its biorthogonalbasis. In this theorem, an arbitrary function fε^(m)S satisfies thefollowing formulae (8) and (9) for a sampling value f_(n) :=f(n).$\begin{matrix}{{f(t)} = {\sum\limits_{n = {- \infty}}^{\infty}\quad {f_{n}{{\,_{\lbrack{\,^{m}S}\rbrack}\varphi}\left( {t - n} \right)}}}} & (8) \\{f_{n} = {\int_{- \infty}^{\infty}{{f(t)}\overset{\_}{{{\,_{\lbrack{{}_{}^{}{}_{}^{}}\rbrack}\varphi}\left( {t - n} \right)}{t}}}}} & (9)\end{matrix}$

where _([) _(^(m)) _(S])φε^(m)S satisfying${{\,_{\lbrack{\,^{m}S}\rbrack}\varphi} \in^{m}S} = {\left( {{1/2}\pi} \right){\int_{- \infty}^{\infty}{\left\{ {\sum\limits_{q = {- \infty}}^{\infty}\quad \left\lbrack {\left( {- 1} \right)^{q}\left( {1 - {q\left( {2{\pi/\omega}} \right)}} \right)} \right\rbrack^{m}} \right\}^{- 1}\quad \times {\exp \left( {{\omega}\quad t} \right)}{\omega}}}}$

on the other hand, _([) _(^(m)) _(S*])φε^(m)S satisfying${\int_{- \infty}^{\infty}{{{\,_{\lbrack{\,^{m}S}\rbrack}\varphi}\left( {t - n} \right)}\overset{\_}{{\,_{\lbrack{{}_{}^{}{}_{}^{}}\rbrack}\varphi}\left( {t - p} \right)}{t}}} = \left\{ \begin{matrix}{1,} & {p = n} \\{0,} & {p \neq n}\end{matrix} \right.$

The formula (8) indicates a function of the fluency transform derivedfrom the sample value, and represents an expansion form, in which thesampling value sequences is a expansion coefficient. On the other hand,the formula (9) indicates a function of the fluency inverse transformderived from the function of the fluency transform, and represents anoperator, which obtains the sampling value sequences from the functionof the fluency transform in the form of a integral transform. Note that,p is an arbitrary integer, and a bar expressed above “φ” in the formula(9) represents a conjugated complex of φ.

Further, in the fluency function space ^(m)S, an orthogonal transform isderived from a fluency transform theorem. The orthogonal transformprovides a generalization of a frequency concept in terms of agreementwith Fourier transform at m=∞, and characterizes a harmonic structure.The theorem satisfies the following two formulae for an arbitraryfunction f∈^(m)S. $\begin{matrix}{{f(t)} = {\left( {{1/2}\pi} \right){\int_{- \pi}^{\pi}{{F(u)}{{\,_{\lbrack{\,^{m}o}\rbrack}\varphi}\left( {t,u} \right)}\quad {u}}}}} & (10) \\{{F(u)} = {\int_{- \infty}^{\infty}{{f(t)}\overset{\_}{{{\,_{\lbrack{\,^{m}o}\rbrack}\varphi}\left( {t,u} \right)}\quad {t}}}}} & (11)\end{matrix}$

where${{\,_{\lbrack{\,^{m}o}\rbrack}\varphi}\left( {t,u} \right)} = {\left( {{1/2}\pi} \right){\int_{- \infty}^{\infty}{\left\{ {\sum\limits_{p = {- \infty}}^{\infty}\quad {\delta \left( {\omega - u + {2\quad \pi \quad p}} \right)}} \right\} \times \left\{ {\sum\limits_{q = {- \infty}}^{\infty}\quad \left( {1 - {q\left( {2{\pi/\omega}} \right)}} \right)^{{- 2}m}} \right\}^{{- 1}/2} \times {\exp \left( {\quad \omega \quad t} \right)}{\omega}}}}$

Note that, φ (t,u) is a function of the fluency function space ^(m)S andis expressed by using a Dirac delta function δ. “u” is an arbitraryvariable.

The formula (10) is designated as a fluency orthogonal transform, andthe formula (11) is designated as a fluency inverse orthogonaltransform.

The fluency orthogonal transform expressed by the formula (10) cantransform discrete sample values to continuous function values.Accordingly, in this embodiment, the fluency orthogonal transform isutilized for expanding the reduced-image data. Namely, the fluencytransform is executed to each pixel of the reduced-image data R, andthen the expanded-image data J is generated on the basis of thecontinuous function values.

Herein, some concrete fluency functions are explained. The order “m” ofthe fluency function space ^(m)S can be expressed as a parameter “m” ofthe fluency function, and when the parameter “m” is set to “1,2, . . . ”in order, the fluency function are represented as shown below.

The most simple function system in the fluency function is obtained bysetting the function “f(t)ε¹S” in the formula (6) to the rectangularfunction, which is represented by “χ(t)” in the formula (5), in place ofthe staircase function, and setting the function gε¹S represented in theformula (7) to the above rectangular function “f(t)” . Namely, thefunction f(t) represented in the formula (6) becomes a rectangularfunction by applying a Delta (δ) function to an input function of theformula (6) in place of the rectangular function expressed by theformula (5). Then, the rectangular function f(t) is utilized for afunction of the convolution integral in place of the function g(τ)represented in the formula (7), when transforming the function space^(m−1)S to the function space ^(m)S.

An input value at the formula (6) becomes the δ function shown in FIG.7A in place of the rectangular function χ(t) represented in the formula(5). The value of the δ function is 1 when variable t is τ, and is 0when variable t is not τ. The fluency function f(t) with the parameterm=1 is represented as following formula, in place of the formula (6).$\begin{matrix}{{f(t)} = {{\sum\limits_{n = {- \infty}}^{\infty}\quad {\chi_{n}{\chi \left( {t - n} \right)}}} = \left\{ {{\begin{matrix}{1,} & {{\tau - {1/2}} < t < {\tau + {1/2}}} \\{0,} & {otherwise}\end{matrix}\quad {f(t)}} \in^{1}S} \right.}} & (12)\end{matrix}$

The fluency function f(t) in the formula (12) becomes a rectangularfunction, as shown in FIG. 7B. Then, the fluency function with theparameter m=2, denoted by “g(t)”, is found by executing the convolutionintegral, on the basis of the rectangular function f(t), as shown in thefollowing formula. $\begin{matrix}{{g(t)} = {{\int_{- \infty}^{\infty}{{f\left( {t - \tau} \right)}{f\quad(\tau)}{\tau}}} = \left\{ {{\begin{matrix}{{t - \tau + 1},} & {{\tau - 1} < t \leq \tau} \\{{{- t} + \tau + 1},} & {\tau < t < {\tau + 1}} \\{0,} & {otherwise}\end{matrix}\quad {g(t)}} \in^{2}S} \right.}} & (13)\end{matrix}$

The fluency function g(t) obtained by the formula (13) is, as shown inFIG. 7C, a triangular function.

When the parameter m is 3, 4, 5 . . . , the convolution integral isexecuted, similarly to the parameter m=2. Namely, based on the fluencyfunction at the parameter “m−1” and the function f(t) represented in theformula (12), the convolution integral is executed, so that the fluencyfunction with the parameter m is generated. For example, when theparameter m is 3, based on the function g(t) obtained by the formula(13) and the function f(t) in the formula (12), the convolution integralis executed, so that a fluency function h(t) shown in FIG. 7D, which issmooth and expressed by a curve, is generated. The fluency function h(t)is expressed as $\begin{matrix}{{h(t)} = {{{\int_{- \infty}^{\infty}{{g\left( {t - \tau} \right)}{f(\tau)}\quad {\tau}\quad {h(t)}}} \in^{3}S} = \left\{ \begin{matrix}{{{- \frac{1}{4}}\left( {t - \tau + 2} \right)^{2}},} & {{\tau - 2} < t < {\tau - \frac{3}{2}}} \\{{{\frac{3}{4}\left( {t - \tau + 1} \right)^{2}} + {\frac{1}{2}\left( {t - \tau + 1} \right)}},} & {{\tau - \frac{3}{2}} \leq t < {\tau - 1}} \\{{{\frac{5}{4}\left( {t - \tau + 1} \right)^{2}} + {\frac{1}{2}\left( {t - \tau + 1} \right)}},} & {{\tau - 1} \leq t < {\tau - \frac{1}{2}}} \\{{{{- \frac{7}{4}}\left( {t - \tau} \right)^{2}} + 1},} & {{\tau - \frac{1}{2}} \leq t < {\tau + \frac{1}{2}}} \\{{{\frac{5}{4}\left( {t - \tau - 1} \right)^{2}} - {\frac{1}{2}\left( {t - \tau - 1} \right)}},} & {{\tau + \frac{1}{2}} \leq t < {\tau + 1}} \\{{{\frac{3}{4}\left( {t - \tau - 1} \right)^{2}} - {\frac{1}{2}\left( {t - \tau - 1} \right)}},} & {{\tau + 1} \leq t < {\tau + \frac{3}{2}}} \\{{{- \frac{1}{4}}\left( {t - \tau - 2} \right)^{2}},} & {{\tau + \frac{3}{2}} \leq t < {\tau + 2}} \\{0,} & {otherwise}\end{matrix} \right.}} & (14)\end{matrix}$

In this way, the fluency function varies with the parameter m. Thefluency functions, shown in FIGS. 7B to 7D, correspond to sampling basesrelated to the fluency function space ^(m)S disclosed in theMathematical Physics Journal described above. In this embodiment, thefluency transform (orthogonal transform) to the reduced-image data R isexecuted on the basis of the fluency function in the case of theparameter m=1, 2 and 3.

However, if the function f(t), obtained by the formula (12) and shown inFIG. 7B, is directly utilized at the convolution integral at theparameter m≧3, the value of the fluency function at t=τ does notcoincide with 1. Accordingly, in this embodiment, normalized functionshown in FIG. 8 is applied to the convolution integral in place of thefunction f(t) shown in FIG. 7B. The normalized function f(t) isnormalized as for an area, which is formed on the basis of the t-axisand the function f(t), such that the area of the function f(t) isalways 1. For example, when the parameter m is 3, the normalizedfunction f(t) is represented by $\begin{matrix}{{f(t)} = \left\{ \begin{matrix}{{- \frac{1}{3}},} & {{\tau - 1} < t \leq {\tau - \frac{1}{2}}} \\{\frac{4}{3},} & {{\tau - \frac{1}{2}} < t \leq {\tau + \frac{1}{2}}} \\{- \frac{1}{3}} & {{\tau + \frac{1}{2}} < t < {\tau + 1}} \\{0,} & {otherwise}\end{matrix} \right.} & (15)\end{matrix}$

Thus, the value of the fluency function is always 1 at t=τ.

FIG. 9 is a view showing the fluency transform for generating theexpanded-image data J. Note that, for ease of explanation, the fluencytransform is executed for only the horizontal direction, or u-axis.

Herein, as one example, three pixels R_(ts−1), R_(ts), R_(ts+1),adjoining each other in the reduced-image data R, are subjected to thefluency transform by the formula (10) along the horizontal direction.Each pixel value of the three pixels R_(ts−1), R_(ts), R_(ts+1) isdifferent respectively. The input function F(u) at the formula (10)corresponds to each pixel value of the three pixels R_(ts−1), R_(ts),R_(ts+1).

When the input function F(u) at the formula (10) is a discrete valuecorresponding to δ function, the output function f(t) in the formula(10) corresponds to the fluency functions shown in FIGS. 7A to 7D.Therefore, by the fluency orthogonal transform, the output function f(t)having continuous values is generated. When the parameter m=1, an outputfunction f_(s), obtained by executing the fluency transform to the pixelvalue R_(ts), is a rectangular function, as shown in FIG. 9. Each pixelvalue of 8 pixels along the horizontal direction (x-axis) in the blockBP, represented by a bar “Q”, is equal.

Then, when the parameter m is 1, the range of the output function f_(s)along the horizontal direction corresponds to the range of thehorizontal direction of the block BP (represented by “L” in FIG. 9). Therange of the output functions f_(s−1), f_(s+1), obtained by executingthe fluency transform to the pixel values R_(ts−1), R_(ts+1), does notoverlap the range of the output function f_(s).

On the other hand, when the parameter m is 2, each of the functionsf_(s−1), f_(s), f_(s+1), obtained by the formula (10) and correspondingto the pixel value R_(ts−1), R_(ts), R_(ts+1) respectively, is atriangular function. In the case of the parameter m=2, each range offunctions f_(s−1), f_(s), f_(s+1) overlaps each other, as the range L ofthe block BP is defined in accordance with the parameter m=1. Thedifference between the range of the function f(t) at the parameter m=1and that of the function f(t) at the parameter m=2 is shown in FIGS. 7Band 7C.

Accordingly, each pixel value, represented by the bar Q, is obtained byadding each value of the functions f_(s−1), f_(s), f_(s+1). Namely, eachvalue of the functions f_(s−1), f_(s), f_(s+1), corresponding to eachposition of 8 pixels in the horizontal direction of the block BP, isadded, so that each pixel value of 8 pixels along the horizontaldirection is obtained. For example, using the second pixel from theright end in the block BP, the pixel value Z3 is obtained by adding thepixel value Z2, which is the value of the function f_(s), and the pixelvalue Z1, which is the value of the function f_(s+1) (See FIG. 9).

As the parameter “m” becomes large, the range of the function f(t) alongthe horizontal direction becomes larger, as shown in FIG. 7D. In thiscase, each pixel value of the block BP, corresponding to the pixelR_(ts) is calculated on the basis of the pixel value R_(ts−1), R_(ts+1)and other adjacent pixels in the reduced-image data R.

After the fluency transform along the horizontal direction is executed,the fluency transform along the vertical direction is executed to theblock BP, so that the block B0′ composed of 8×8 pixels is generated.Namely, when executing the fluency transform along the horizontaldirection and the vertical direction for each pixel of the reduced-imagedata R in order, the expanded-image data J is generated. Each pixelvalue of the expanded-image data J depends upon the parameter m, and theparameter m is the mode of the fluency transform.

With reference to FIGS. 10 to 13, expansion processing by the fluencytransform is explained. In this embodiment, the parameter m is one of 1to 3, and adjacent pixels to the pixel R_(ts), necessary for findingeach pixel of the block BP, are represented by R_(ts−2), R_(ts−1),R_(ts+1), R_(ts+2), R_(ts−2s), R_(t−1s), R_(t+1s), R_(t+2s). Note that,values of the pixels R_(ts−2), R_(ts−1), R_(ts+1), R_(ts+2), R_(t−2s),R_(t−1s), R_(t+1s), R_(t+2s) are also expressed by “R_(ts−2), R_(ts−1),R_(ts+1), R_(ts+2), R_(t−2s), R_(t−1s), R_(t+1s), R_(t+2s)”.

FIG. 10 is a view showing the fluency transform along the horizontaldirection (width direction). The positions of 8 pixels in the block BPare expressed by “0, 1, 2 . . . 7” in order.

Firstly, the pixel R_(ts), is arranged at the center of the generatedblock BP (between the third and fourth position), and then the fluencytransform at the parameter m (=1, 2 or 3) is executed along thehorizontal direction. Similar to the pixel R_(ts), the pixels R_(ts−2),R_(ts−1), R_(ts+1), R_(ts+2), R_(t−2s), R_(t−1s), R_(t+1s), R_(t+2s) aresubject to the fluency transform in the horizontal direction, at thecenter of each block generated by the fluency transform. 8 pixels,represented by “I0, I1, I2, I3, I4, I5, I6, I7” in FIG. 10, are based onthe pixels R_(ts−2), R_(ts−1), R_(ts), R_(ts+1), R_(ts+2).

FIG. 11 is a view showing a table T1 indicating 8 pixel values along thehorizontal direction in the block BP obtained by the fluency transform.In the table T1, pixel values corresponding to the parameter 1, 2 and 3respectively, are shown, and each pixel value is obtained on the basisof the formula (10). Herein, pixel values along the horizontal directionare also represented by “I0, I1, I2, . . . I7”, which are identical withthe 8 pixels along the horizontal direction.

For example, when the parameter m is 1, all of the pixel values I0 to I7coincide with the pixel value R_(ts). As described above using FIG. 9,the pixel values R_(ts−2), R_(ts−1), R_(ts+1), R_(ts+2) except for thepixel value R_(ts) in the reduced-image data R do not affect thegeneration of the pixel values I0 to I7 when the parameter m=1. On theother hand, when the parameter m is 2, the pixel values I0 to I7 aregenerated on the basis of the pixel values R_(ts−1), R_(ts), R_(ts+1)and the addition of the value of the function f_(s−1), f_(s), f_(s+1) atthe formula (10), as shown in FIG. 9. Namely, the pixel values I0 to I7are respectively generated by adding each value of the functionsf_(s−1), f_(s), f_(s+1), which correspond to the pixel position. Whenthe parameter m is 3, the pixel values I0 to I7 are obtained on thebasis of the pixel values R_(ts−2), R_(ts−1), R_(ts), R_(ts+1),R_(ts+2). After the fluency transform in the horizontal direction isexecuted, the fluency transform in the vertical direction (y-axis) isexecuted to the block BP composed of 8 pixels I0 to I7.

FIG. 12 is a view showing the fluency transform along the verticaldirection. Similarly to the horizontal direction, the positions of 8pixels along the vertical direction are expressed by “0, 1, 2, . . . 7”in order.

When the fluency transform along the horizontal direction is executed toeach pixel of the reduced-image data R, blocks FP, GP, HP, KP,corresponding to the pixels R_(t−2s), R_(t−1s), R_(t+1s), R_(t+2s)respectively, are obtained. Each pixel of the blocks,FP, GP, HP, KP isdesignated by “f0, f1, . . . f7”, “g0, g1, . . . g7”, “h0, h1, . . .h7”, “k0, k1, . . . k7”, respectively. Note that, Each pixel of theblocks FP, GP, HP, KP is obtained on the basis of other adjacent pixelsin the reduced-image data R in addition to R_(ts−2), R_(ts−1), R_(ts),R_(ts+1), R_(ts+2). For example, “f0, f1, f7” of the block FP isobtained by “R_(t−2s−2), R_(t−2s−1), R_(t−2s), R_(t−2s), R_(t−2s+2).

The 8 pixels “I0, I1, I2, . . . I7” are subjected to the fluencytransform along the vertical direction at the center of the block BP(between the third position and the fourth position), on the basis ofthe formula (10). The other pixels “f0, f1, . . . f7”, “g0, g1, . . .g7”, “h0, h1, . . . h7”, “k0, k1, . . . k7” are also subjected to thefluency transform along the vertical direction respectively, similar tothe 8 pixels I0 to I7. Then, 8×8 pixels of the block B0′ are obtained byadding each value of: (a) The output values f(t) (at the formula (10))based on the pixels “f0, f1, . . . f7” corresponding to the pixelpositions “0, 1, 2, . . . 7”, (b) The output functions f(t) based on thepixels “g0, g1, . . . g7” corresponding to the pixel positions “0, 1, 2,. . . 7”, (c) The output values f(t) based on the pixels “I0, I1, . . .I7” corresponding to the pixel positions “0, 1, 2, . . . 7”, (d) Theoutput values f(t) based on the pixels “h0, h1, . . . h7” correspondingto the pixel positions “0, 1, 2, . . . 7”, (e) The output values f(t)based on the pixels “k0, k1, . . . k7” corresponding to the pixelpositions “0, 1, 2, . . . 7”. The pixel value of the block B0′ isdenoted by I′_(yx) (0≦x≦7, 0≦y≦7), as described above.

FIG. 13 is a view showing a table T2 representing 8 pixel values I′_(y7)(0≦y≦7), shown by oblique line in FIG. 12. When the parameter m is 1,all of the pixel values I′_(y7) are I7, when the parameter m is 2, thepixel values I′_(y7) are obtained on the basis of the pixel values I7,g7, h7. When the parameter m is 3, the pixel values I′_(y7) are obtainedon the basis of the pixel values f7, g7, I7, h7, k7. Other pixels I_(yx)(0≦x≦6, 0≦y≦7) are also obtained on the basis of the pixels fx, gx, Ix,hx, kx (x=0 to 6). In this case, the pixel values I_(yx) (0≦x≦6, 0≦y≦7)are obtained by substituting fx, gx, Ix, hx, kx (x=0 to 6) for f7, g7,I7, h7, k7 represented in the table T2.

In this way, the block B0′ corresponding to the block B0 composing theoriginal image data P is generated by executing the fluency transformalong the horizontal and the vertical direction. The fluency transformis executed for each pixel of the reduced-image data R in order, so thatall of the block B′ is generated, namely, the expanded-image data J isobtained.

FIG. 14 is a view showing a flowchart of the process for determining theoptimum mode. In FIG. 14, the optimum mode for the block B0′ isdetermined.

In Step 101, the parameter m is set to 1. Then, in Step 102, the blockB0′ composed of 8×8 pixels is generated by the fluency transform, andthen the code-length E(m) at parameter m=1, represented by E(1), iscalculated. The code-length E(m) is equal in value to a bit length ofthe encoded bit data of one block.

The calculation of the code-length is partially different from that ofthe Huffman coding. Firstly, similarly to the Huffman coding, thedifferential DCT coefficient matrix is zigzag scanned. Then, thedifferential DCT coefficients C_(ji), which is not “0”, are categorizedusing the categorization table so that a length of the additional bits(not additional bits) are obtained. Further, when the zero run length Zis obtained from the differential DCT coefficients C_(ji) of “0”, basedon the category number H and the zero run length Z, a length of theHuffman code-word (not the Huffman code-word) is obtained using theHuffman table. Then, the length of the additional bits and the length ofthe Huffman code-word is added sequentially, so that the code-length ofthe encoded bit data is calculated without generating the encoded bitdata. Note that, the above calculation is conventionally well known.

Furthermore, at Step 102, an optimum parameter “md”, namely, the optimummode, is set to 1, and then a minimum code-length E_(min) is set toE(1).

In Step 103, the parameter m is incremented by 1, and the code-lengthE(m) is calculated on the basis of the incremented parameter m.

In Step 104, it is determined whether the code-length E(m) obtained atStep 103 is smaller than the minimum code-length E_(min). When it isdetermined that the code-length E(m) is smaller than the minimumcode-length E_(min), the process goes to Step 105, wherein thecode-length E(m) is newly set to the minimum code-length E_(min), andthe parameter m is set to the optimum parameter md. After Step 105 isexecuted, the process goes to Step 106. On the other hand, when it isdetermined that the code-length E(m) is not smaller than the minimumcode-length E_(min) at Step 104, the process skips Step 105 and directlygoes to Step 106.

In Step 106, it is determined whether the parameter m is 3. When it isdetermined that the parameter m is 3, the process for determining theoptimum mode is terminated. On the other hand, when it is determinedthat the parameter m is not 3, the process returns to Step 103.

Steps 103 to 106 are repeatedly executed, thus the optimum mode md,which makes the code-length E(m) minimum, is determined among 1 to 3.

The process is executed to all the other blocks in the expanded-imagedata J, similarly to the block B0′, so the optimum mode is determinedfor each block.

In this way, in this embodiment, the original image data P istransformed to the reduced-image data R, and is recorded in therecording medium M in the image compression apparatus 10. Further, thereduced-image data R is subjected to the fluency transform so as togenerate the expanded-image data J, and the differential value dataindicating the difference between the original image data P and theexpanded-image data J is obtained in the image compression apparatus 10.The differential value data is subjected to the Huffman coding, so thatencoded bit data is obtained and recorded in the recording medium M. Inthe image expansion apparatus 20, the recorded reduced-image data R istransformed to the expanded-image data J, and the encoded bit data istransformed to the differential value data. Then, the original imagedata P is restored on the basis of the differential value data and theexpanded-image data J. As the expanded-image data J is generated anddifferential value data is obtained in the image compression apparatus10 in advance, the original image data P is perfectly restored in theimage expansion apparatus 20. Then, as the differential value data isHuffman encoded, the bit length (the amount of bit data) recorded in therecording medium M is decreased. Hence, the original image data P isefficiently compressed, while maintaining the picture quality.

As the fluency transform is performed in the expansion processing, thedifference between each pixel of the original image data P and thecorresponding pixel of the expanded-image data J is smaller incomparison with that of the linear interpolation. Namely, each pixelvalue of the difference value data is relatively small. Accordingly, thebit length of the encoded bit data is decreased so that the originalimage data P is efficiently compressed.

The optimum mode, which makes the code-length minimum, is determinedamong the parameter m=1 to 3, thus the bit length of the encoded bitdata is suppressed. Furthermore, as the optimum mode is determined ineach block in the expanded-image data J, the bit length of the encodedbit data is further decreased.

Note that, the differential DCT coefficient data may be quantized.Namely, a quantization processor may be provided between the DCTprocessor 15 and the Huffman coding processor 18. In this case, thedifferential DCT coefficient data is quantized, and the code-lengthnecessary for the Huffman coding is calculated on the basis of thedifferential DCT coefficient data. Then, the inverse quantization isexecuted to the differential DCT coefficient data in the image expansionapparatus 20.

With reference to the orthogonal transform, a Hadamard transform may beapplied in place of the DCT processing executed in the DCT processor 15.In this case, an inverse Hadamard transform is executed in the imageexpansion apparatus 20. Further, an Arithmetic coding, which is one ofthe Entropy coding, may be applied in place of the Huffman coding in theimage compression apparatus 10. In this case, an Arithmetic decoding isexecuted in the image expansion apparatus 20.

The range of the parameter m is not restricted to 1 to 3, and may be setto an arbitrary range (for example, 1 to 5).

When obtaining the reduced-image data R, other methods, such as a downsampling method transforming an image-resolution, can be applied inplace of the calculation of the average value. Further, in place of thedigital still camera, a program executing the image compression andexpansion described above may be stored in a memory in a computer systemso as to compress and expand the image data recorded in the computersystem.

Finally, it will be understood by those skilled in the art that theforegoing description is of preferred embodiments of the device, andthat various changes and modifications may be made to the presentinvention without departing from the spirit and scope thereof.

The present disclosure relates to subject matters contained in JapanesePatent Application No. 11-257095 (filed on Sep. 10, 1999) which isexpressly incorporated herein, by reference, in its entirety.

What is claimed is:
 1. A compression apparatus for compressing imagedata comprising: a reduced-image generating processor that transformsoriginal image data partitioned into first blocks, each of which iscomposed of a plurality of pixels, to reduced-image data composed of asmaller number of pixels than that of said original image data; afluency transform processor that applies a fluency transform to saidreduced-image data so as to generate expanded-image data partitionedinto second blocks corresponding to said first blocks, said fluencytransform having a plurality of modes; a differential value calculatingprocessor that obtains differential value data indicating a differencebetween said original image data and said expanded-image data; anorthogonal transform processor that obtains orthogonal transformcoefficient data by applying an orthogonal transform to saiddifferential value data; a mode setting processor that selects one modefrom said plurality of modes, said orthogonal transform coefficient databeing generated in accordance with said selected mode; a code-lengthcalculating processor that calculates a code-length corresponding to abit length of Entropy-encoded bit data obtained by an Entropy coding tosaid orthogonal transform coefficient data, said code-length calculatingprocessor calculating said code-length in each of said plurality ofmodes; an optimum mode determining processor that determines an optimummode, by which said code-length becomes minimum, from said plurality ofmodes; and an optimum Entropy coding processor that obtains saidEntropy-encoded bit data by applying said Entropy coding to saidorthogonal transform coefficient data in accordance with said optimummode.
 2. The compression apparatus according to claim 1, furthercomprising a recording medium, said reduced-image data, saidEntropy-encoded bit data and said optimum mode being recorded in saidrecording medium.
 3. The compression apparatus according to claim 1,wherein each of said first blocks and each of said second blocks arecomposed of 8×8 (=64) pixels respectively.
 4. The compression apparatusaccording to claim 1, wherein said reduced-image generating processorgenerates said reduced-image data composed of average pixels, obtainedby finding an average value of said plurality of pixels in each of saidfirst blocks.
 5. The compression apparatus according to claim 1, whereinsaid fluency transform processor generates said expanded-image data byapplying the fluency transform, which is expressed by the followingformula, to each pixel of said reduced-image data${f(t)} = {\left( {{1/2}\pi} \right){\sum\limits_{- \pi}^{\pi}\quad {{F(u)}{{\,_{\lbrack{\,^{m}o}\rbrack}\varphi}\left( {t,u} \right)}{u}}}}$

note that,${{\,_{\lbrack{\,^{m}o}\rbrack}\varphi}\left( {t,u} \right)} = {\left( {{1/2}\pi} \right){\int_{- \infty}^{\infty}{\left\{ {\sum\limits_{p = {- \infty}}^{\infty}\quad {\delta \left( {\omega - u + {2\pi \quad p}} \right)}} \right\} \times \left\{ {\sum\limits_{q = {- \infty}}^{\infty}\quad \left( {1 - {q\left( {2{\pi/\omega}} \right)}} \right)^{{- 2}m}} \right\}^{{- 1}/2}\quad \times \exp \quad \left( {{\omega}\quad t} \right){\omega}}}}$

where F(u) corresponds to each pixel value of said reduced-image data,f(t) is output values of the fluency transform, φ (t,u) is the fluencyfunction defined by a fluency function space ^(m)S, and m (=1,2,3, . . .) is a parameter indicating a differentiability, said parameter mcorresponding to said plurality of modes.
 6. The compression apparatusaccording to claim 5, wherein said fluency transform processor appliesthe fluency transform in a state such that each pixel of saidreduced-image data are arranged at a center position of each of saidsecond blocks, and generates said expanded-image data composed of saidplurality of pixels by adding said output values f(t) corresponding topixel positions in said second blocks.
 7. The compression apparatusaccording to claim 6, wherein said plurality of pixels in said originalimage data is arranged in a matrix, said fluency transform processorapplies the fluency transform along a width direction to each pixel ofsaid reduced-image data so that pixels aligned along the width directionare generated, and then applies the fluency transform along a lengthdirection to said pixels aligned along said width direction such thatsaid plurality of pixels in each of said second blocks is generated. 8.The compression apparatus according to claim 1, wherein said optimummode is determined in each of said second blocks.
 9. The compressionapparatus according to claim 1, wherein said orthogonal transformprocessor applies a DCT (Discrete Cosine Transform) processing to saiddifferential value data such that a differential DCT coefficient data isobtained.
 10. The compression apparatus according to claim 1, whereinsaid Entropy coding processor applies a Huffman coding to saidorthogonal transform coefficient data so that Huffman-encoded bit datais obtained, said code-length calculating processor calculating saidcode-length corresponding to said Huffman-encoded bit data, said optimummode determining processor determining a mode, which makes saidcode-length minimum, among said plurality of modes as said optimum mode.11. An expansion apparatus for expanding said reduced-image datarecorded in said recording medium by said compression apparatus in claim2, said expansion apparatus comprising: a data reading processor thatreads said reduced-image data, said Entropy-encoded bit data and saidoptimum mode recorded in said recording medium; an optimum mode settingprocessor that sets said optimum mode among said plurality of modes; anexpanded-image generating processor that applies said fluency transformaccording to said optimum mode to said reduced-image data such that saidexpanded-image data is obtained; an Entropy decoding processor thatrestores said orthogonal transform coefficient data by applying anEntropy-decoding to said Entropy-encoded bit data; an inverse orthogonaltransform processor that restores said differential value data byapplying an inverse orthogonal transform to said orthogonal transformcoefficient data; an original image data restoring processor thatrestores said original image data on the basis of said expanded-imagedata and said differential value data.
 12. The expansion apparatusaccording to claim 11, wherein said Entropy-encoded bit data isHuffman-encoded bit data, said Entropy decoding processor applying aHuffman decoding to said Huffman-encoded bit data.
 13. The expansionapparatus according to claim 11, wherein said orthogonal transformcoefficient data is a differential DCT coefficient data, said inverseorthogonal transform processor applying an inverse DCT processing tosaid differential DCT coefficient data.
 14. A compression method forcompressing image data comprising: transforming original image datapartitioned into first blocks, each of which is composed of a pluralityof pixels, to reduced-image data composed of a smaller number of pixelsthan that of said original image data; applying a fluency transform tosaid reduced-image data so as to generate expanded-image datapartitioned into second blocks corresponding to said first blocks, saidfluency transform having a plurality of modes; obtaining differentialvalue data indicating a difference between said original image data andsaid expanded-image data; obtaining orthogonal transform coefficientdata by applying an orthogonal transform to said differential valuedata; selecting one mode from said plurality of modes, said orthogonaltransform coefficient data being generated in accordance with saidselected mode; calculating a code-length corresponding to a bit lengthof Entropy-encoded bit data obtained by an Entropy coding to saidorthogonal transform coefficient data, said code-length being calculatedin each of said plurality of modes; determining an optimum mode, bywhich said code-length becomes minimum, among said plurality of modes;and obtaining said Entropy-encoded bit data by applying said Entropycoding based on said optimum mode to said orthogonal transformcoefficient data.
 15. An expansion method for expanding saidreduced-image data obtained by said compression method in claim 14, saidexpansion method comprising: reading said reduced-image data, saidEntropy-encoded bit data and said optimum mode recorded in a recordingmedium; setting said optimum mode among said plurality of modes;applying said fluency transform according to said optimum mode to saidreduced-image data such that said expanded-image data is obtained;restoring said orthogonal transform coefficient data by applying anEntropy-decoding to said Entropy-encoded bit data; restoring saiddifferential value data by applying an inverse orthogonal transform tosaid orthogonal transform coefficient data; restoring said originalimage data on the basis of said expanded-image data and saiddifferential value data.
 16. A memory medium that stores a program forcompressing image data, said program comprising: transforming originalimage data partitioned into first blocks, each of which is composed of aplurality of pixels, to reduced-image data composed of a smaller numberof pixels than that of said original image data; applying a fluencytransform to said reduced-image data so as to generate expanded-imagedata partitioned into second blocks corresponding to said first blocks,said fluency transform having a plurality of modes; obtainingdifferential value data indicating a difference between said originalimage data and said expanded-image data; obtaining orthogonal transformcoefficient data by applying an orthogonal transform to saiddifferential value data; selecting one mode from said plurality ofmodes, said orthogonal transform coefficient data being generated inaccordance with said selected mode; calculating a code-lengthcorresponding to a bit length of an Entropy-encoded bit data obtained byan Entropy coding to said orthogonal transform coefficient data, saidcode-length being calculated in each of said plurality of modes;determining an optimum mode, by which said code-length becomes minimum,among said plurality of modes; and obtaining said Entropy-encoded bitdata by applying said Entropy coding, based on said optimum mode, tosaid orthogonal transform coefficient data.
 17. A memory medium thatstores a program for expanding said reduced-image data obtained by saidprogram for compressing image data in claim 16, said program forexpanding comprising: reading said reduced-image data, saidEntropy-encoded bit data and said optimum mode recorded in a recordingmedium; setting said optimum mode from said plurality of modes; applyingsaid fluency transform according to said optimum mode to saidreduced-image data such that said expanded-image data is obtained;restoring said orthogonal transform coefficient data by applying anEntropy-decoding to said Entropy-encoded bit data; restoring saiddifferential value data by applying an inverse orthogonal transform tosaid orthogonal transform coefficient data; restoring said originalimage data on the basis of said expanded-image data and saiddifferential value data.